Note that, in real applications, the fault features are not strictly group-sparse periodic. In other words, the real fault frequencies have a variation from the calculated frequencies (up to 1-2%), i.e., there are not strict periods \(T_1\)  and \(T_2\)  [1]}.
In such cases, the proposed approach can still work owing to the overlapping of group structure, which will be shown in the following sections.
The proposed approach is not an improvement of the cutting edge techniques developed in the cyclostationary framework, instead, it is an alternative to these techniques.
A work closely related to the proposed approach is the periodic overlapping group sparsity (POGS) problem [2]},
which assumes only one repetitive group-sparse component is present in the vibration signal.
Notice that although the observed signal is modeled with two components,
the proposed method also works when there exist only one fault component.
Therefore, the proposed method generalizes POGS, and is useful
when there exists multiple components.
